As argued in [1]}, the assumed exchangeability of the offspring distribution in Cannings' models
ensures that the partition-valued ancestry process evolves, in distribution,
through sequential “putting balls into boxes” steps.
In fact it is from this last property from which the Markovicity and exchangeability of the ancestry process stem.
In this work we thus remove the exchangeability condition on the offspring distribution of Cannings' models,
and instead model neutral evolution by removing any form of inheritance, letting children choose parents
in an exchangeable way. In other words, we allow individuals
in any given generation to have different offspring distributions,
but impose that these differences in reproductive success
stem from non-heritable traits. An example of this would be a population whose individuals randomly explore different
ecological niches in every new generation, resulting in non-heritable asymmetries in reproductive success. Formally, the models we consider in this work consist of a population with constant size \(N\)  evolving in discrete time.
Letting \(\xi ^{(t)}_N\left(\xi _{1,N}^{(t)},\cdots ,\xi _{N,N}^{(t)}\right)\)  be the offspring sizes of each parent in generation
\(t\in \mathbb {N}\) , the
framework makes the following assumptions.
